$ A = \left[\begin{array}{rr}1 & 3 \\ 0 & -2 \\ 5 & 0\end{array}\right]$ $ C = \left[\begin{array}{rr}-1 & -2 \\ 0 & 3\end{array}\right]$ What is $ A C$ ?
Because $ A$ has dimensions $(3\times2)$ and $ C$ has dimensions $(2\times2)$ , the answer matrix will have dimensions $(3\times2)$ $ A C = \left[\begin{array}{rr}{1} & {3} \\ {0} & {-2} \\ \color{gray}{5} & \color{gray}{0}\end{array}\right] \left[\begin{array}{rr}{-1} & \color{#DF0030}{-2} \\ {0} & \color{#DF0030}{3}\end{array}\right] = \left[\begin{array}{rr}? & ? \\ ? & ? \\ ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ A$ , with the corresponding elements in column $j$ of the second matrix, $ C$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ A$ with the first element in ${\text{column }1}$ of $ C$ , then multiply the second element in ${\text{row }1}$ of $ A$ with the second element in ${\text{column }1}$ of $ C$ , and so on. Add the products together. $ \left[\begin{array}{rr}{1}\cdot{-1}+{3}\cdot{0} & ? \\ ? & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ A$ with the corresponding elements in ${\text{column }1}$ of $ C$ and add the products together. $ \left[\begin{array}{rr}{1}\cdot{-1}+{3}\cdot{0} & ? \\ {0}\cdot{-1}+{-2}\cdot{0} & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ A$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ C$ and add the products together. $ \left[\begin{array}{rr}{1}\cdot{-1}+{3}\cdot{0} & {1}\cdot\color{#DF0030}{-2}+{3}\cdot\color{#DF0030}{3} \\ {0}\cdot{-1}+{-2}\cdot{0} & ? \\ ? & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rr}{1}\cdot{-1}+{3}\cdot{0} & {1}\cdot\color{#DF0030}{-2}+{3}\cdot\color{#DF0030}{3} \\ {0}\cdot{-1}+{-2}\cdot{0} & {0}\cdot\color{#DF0030}{-2}+{-2}\cdot\color{#DF0030}{3} \\ \color{gray}{5}\cdot{-1}+\color{gray}{0}\cdot{0} & \color{gray}{5}\cdot\color{#DF0030}{-2}+\color{gray}{0}\cdot\color{#DF0030}{3}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rr}-1 & 7 \\ 0 & -6 \\ -5 & -10\end{array}\right] $